Sunday, April 29, 2012

Stress Analysis of Piping

The analysis of piping under pressure, weight and thennal expansion is complex. This complexity can be understood by knovledge of Principal Axis System.

Stress is considered as the ratio of Force to Area. To find the stress in the small element, say cube of a piece of pipe, construct a three-dimensional, mutually perpendicular principal axis system with each axis perpendicular to the face of the cube it intersects.

Each force, acting on the cube can be resolved into force components, acting along each of the axis. Each force, acting on the face of the cube divided by area of the cube face is called the principal stress.

The principal stress acting along the centerline of the pipe is called Longitudinal principal stress. This stress is caused by longitudinal bending, axial force loading or pressure.

Radial principal stress acts on a line from a radial line from center of pipe through the pipe wall. This stress is compressive stress acting on pipe inside diameter caused by internal pressure or a tensile stress caused by vacuum pressure.

Circumferential principal stress, some times called Hoop or tangential stress, acts along the circumference of the pipe. This stress tends to open-up the pipe wall and is caused by internal pressure.

When two or more principal stresses act at a point on a pipe, a shear stress will be generated.

Longitudinal Principal stress, LPS = PD/4T
Circumferential Principal stress, CPS (Hoop) = PD/2T
Radial Principal stress, RPS = P

Failure Theories

The Code presents equations for detennining the stress levels in a piping system & provides stress limits for comparison. These theories are maximum principal stress failure theory & maximum shear stress failure theory.

The maximum principal stress failure theory states that when anyone of the three mutually perpendicular principal stresses exceed the yield strength of the material at temperature, failure will occur.

The maximum shear failure theory states that when the maximum shear stress (arithmetic average of largest minus smallest principal stresses) exceeds one-half the yield strength of the material at temperature, failure will occur.

Stress Types

The B31.3 Code provides design guidance for primary & secondary stresses. The basic characteristic of a primary stress is that it is not self-limitinI!. ..\5 long as the load is applied, the stress will be present & will not diminish \\~th time or as defonnation takes place. The failure mode of a primary stress. is gross defonnation progressing to rupture. Examples of a primary stress are circumferential stresses due to internal pressure & longitudinal bending stresses due to gravity. The basic characteristic of a secondary stress is that it is self- limiting. The stress will diminish with time and strain. The failure mode of a secondary stress is small crack leading to leakage. Secondary stresses are due to cyclic thermal expansion and contraction.

Wall thickness for Internal Pressure

Calculate the adequate pipe wall thickness for a given material and design conditions, as follows:

1. Calculate pressure design thickness “t” with formula

t = P x D I [2 (SE+PY)] ... ... ... ... Eq. (3a)

P = internal design gauge pressure
D = pipe outside diameter
S = Shallowable stress from Appendix A-I, B31.3
E = Welding Quality factor
Y = stress-temperature compensating factor from table 304.1.1

2. Add the mechanical corrosion/erosion allowances “c” to obtain

t(m) = t+c

3. Add mill tolerance to t(m) to select next commercially available schedule wall thickness.

WaIt thickness for External Pressure:(REA) (Read with Appendix-A)

The required minimum thickness of a pipe under external pressure is determined as detailed out in ASME Section VIII Division I Para UG-28 for Do/t > 10 as follows:


A = factor determined from 5-UGO-28.0
B = factor determined from the applicable material chart for maximum design metal temperature.
Do = Outside diameter of pipe in inches.
E = Modulus of elasticity at design temperature.
L = Total length of pipe, inches P = External design pressure, psi
Pa = Calculated value of the maximum allowable external working pressure for the assumed value of “t”,
psi T = minimum required thickness of pipe, inches


Assume a value for “t” & determine ratios LIDo & Dolt. If LIDo > 50, assume 50 & LIDo < 0.5, assume 0.5.

1. Enter figure 5-UGO-28.0, Appendix 5 at the value of LIDo.

2. Move vertically to the graphed line on the Dolt graph for the value of UDo. There are several graphs in Section VIII. Select the graph for the material under consideration. From the point of intersection, move horizontally to the left to determine the value of factor A.

3. Enter the value of A in the applicable material graph & move vertically to an intersection with the line. Where the value of A falls to the right of the line, assume an intersection with the horizontal projection of the upper end ofline. For value of A falling to the left of the line refer 5 below.

4. Move horizontally from the intersection to the left to find the value of factor B. Calculate the value of maximum allowable external pressure using following formula: Pa = 4B / 3(Do/t)

5. For tl e values of A fal1ing to the left of the line mentioned in 3 above, calculate Pausing following formula: Pa = 2AE / 3(Do/t)

6. If Pa < P, select a larger value of “t” and repeat the entire procedure till Pa 2: P.

Design of Miters ( Read with Appendix -B)

Miter bends have pressure limitation, as calculated by equations (4a, 4b & 4c of 304.2.3 of B31.3) A miter is defined as an angle off-set greater than three degrees. Multiple miters, whose miter cut angle is less than 22.5 degrees are limited to a pressure that will generate hoop stresses not to exceed 50% of the yield strength of the material at temperature. This is done by restricting the maximum pressure to the lesser value as calculated by equation (4a) or (4b) in the code. Single miters or miters whose bend angle is greater than 22.5 degrees are limited to hoop stresses of 20% of the material: ield strength at temperature by equation (4c).

In all the equations above, T is purchase order thickness i.e. Nominal thickness less mill tolerance.

Design of Branch Connections (Read with Appendix - C)

The amount of required pressure reinforcement is determined by performing area replacement calculations (304.3.3). Area replacement calculations are not required for unlisted tee intersections, provided the tee component meets at least one of the following requirements:

1. Duplicating a successful operating system.

2. Experimental stress analysis.

3. Proof test.

4. Detailed finite element stress analysis.

The B3I.3 procedures for replacement calculations are valid for the conditions:

1. The center line of the branch pipe must intersect the center line of the run.

Intersections that do require reinforcement calculations (those that ale not qualified by ¶ 304.7.2 or by being listed), are qualified by summing all the integral metal around the intersection, (within a prescribed boundary), (reinforcementare area) that is beyond that required to contain pressure and comparing that sum to the metal area removed to make the intersection.

Referring fig. 304.3.3 Branch connection nomenclature,

Blind Flanges

Blind flanges are used to stop the flow of content of piping. They are exposed to full longitudinal pressure force. B31.3 provides an equation to calculate the thickness of blind as follows:

Flexibility Analysis Of Piping Systems

The safety of a piping system subjected to a temperature change and resulting thermal displacement is determined by a flexibility analysis to insure against following:

1. Overstrain of piping components

2. Overstrain of supporting structures

3. Leakage at joints and

4. Over strain of connecting equipment without material waste.

Compliance with B31.3 Code flexibility analysis is a requirement of most petroleum and chemical plant piping installations. The code places the onus of this analysis on the designer and holds the designer responsible to the owner for assuring that all the engineering design complies with the requirements of the Code.

The code is clear as to which piping systems require an analysis, all systems require an analysis with the exception of the following. (319.4.1)

1. Those that are duplicates of successfully operating installations.

2. Those that can be judged adequate by comparison with previously analysed systems and

3. Systems of uniform size that have no more than two anchor points, no intermediate restraints and fall within limitation of the following equation.


D = Outside diameter of the pipe, in. (mm.)
Y = Resultant total displacement strains, in (mm), to be absorbed by the piping system.
L = Developed length of piping between anchors, ft (m).
U = Anchor distance, straight line between anchors, ft (m).
K1 = 0.03 for U.S. customary units (208.3 for SI Units).

The allowable stress range, SA [302.3.4 (d)] is the stress limit for those stresses that are repeated and cyclic in nature. It is the allowable stress to be compared to the calculated displacement stress range, SE (319.4.4).

The allowable stress range is presented by two equations.

Equation (1a):

SA = f (1.25 SC + 0.25 Sh).

And equation (1b):

SA = f [1.25 (SC + Sh) – S1]

Sc and Sh are the basic allowable stresses for the cold and hot conditions. Their values are found in B31.3 appendix A table A1. For cryogenic or cold pipe service, SC is taken at operating temperature and Sh is taken at the installed temperature. ‘f’ is the stress range reduction factor presented in B31.3 table 302.3.5 or equation (1c). SL is the longitudinal stress in the sustained load condition.

Displacement Stress Range

The displacement stress range SE is the calculated range of secondary stress a piping system will generate when subjected to thermal expansion or contraction. Pressure and weight i.e. primary stresses are not considered in this evaluation. This value is compared with allowable stress range, SA. The B31.3 equation for the displacement stress range is

Cold Spring

Cold spring in a piping system is the intentional deformation of the piping for the purpose of reducing pipe end reactions on supports and equipment. This deformation is introduced during fabrication and erection by cutting the pipe length long or short, depending on the expected thermal expansion. Pining systems operating above the installed temperature would be cold sprung by shortening the pipe length by an amount equal to or less than the expected thermal expansion. The thermal reaction is reduced to a lower value Rm by the equation.

Rm = R ( 1 – 0.66 C) Em/En

R = reaction force from thermal analysis.
Em = modulus of elasticity at maximum temperature
En = modulus of elasticity at operating temperature
C = Cold spring factor varying from 0 ( no cold spring) to 1.0 (100% cold spring.)

Cold spring is the process of offsetting or preloading the piping system with displacement loads i.e. cutting short or long the pipe between two anchors.

Occasional Load Stresses:

Occasional load stresses in piping systems are the some of those stresses caused by loads such as relief valve discharge, wind and earthquake. These are calculated considering the:

a. Pipe deflection caused by wind load acting as a horizontal constant pressure on the outside surface of pipe 


b. Pipe deflection caused by earthquake loads, acting as a horizontal or vertical acceleration of the mass or weight.

The wind and ear thquake loads need not be considered as acting simultaneously. The allowable loads for occasional loads, SOL, summed up with the stresses due to sustained loads, SL, is 1.33 Sh i.e. SOL + SL £ 1.33 Sh.

Maximum Span for Piping:

The empirical formula for calculating maximum span is


L = span in metres
S = safe stress in Kg/cm2. (i.e. longitudinal stress due to internal pressure.)
Z = Pipe section modulus in cm3.
W = uniformly distributed load in Kg/m.

The uniformly distributed load W = weight of pipe + weight of medium +weight of insulation ( all per m length.)

Thermal Expansion Analysis:

Simplified Analysis

Simplified analysis is based on guided cantilever method. Guided cantilever is a cantilever restrained by guides at free end in such a way that its free end will not rotate when deflected direction perpendicular to longitudinal axis. The guides themselves displace along with free end. In this case, bending moment M is given by

M = 6EI/ L2

M = Bending moment
E = Modulus of elasticity
I = Moment of Inertia
= Deflection perpendicular to axis
L = Length of span

Corresponding stress S = iM/Z, I = Stress intensification factor Analysis of Thermal Displacements and Expansion Movements

Three simple rules are quite handy in this analysis.

1. For a coplanar piping, thermal movement of pipe perpendicular to the axis depends upon to…. length of piping irrespective of piping routing.

Expansion at point B in both the cases is same.

2. For a vertical stretch of piping of length L, if two anchored horizontal segments of lengths m and are connected then, the nil-displacement point divides length l in the ration m : n.

Also, the vertical expansion is in the ratio of m : n.

3. Displacement triangle rule for a pipe: If ∂ is displacement of one end ‘y’ with respect to other ‘a’, then displacement at any point will be such that

Slope for Piping

To prevent pocketing of water or condensate at low point of the pipe, the pipe must be sloped such that Slope = 4∂ where ∂ is the deflection of pipe.

Expansion Joints:

Expansion joints are used for following reasons:

1. Reduce expansion stresses
2. Reduce piping reactions on connected equipment
3. Reduce pressure drop in system by avoiding long expansion loops
4. Isolate mechanical vibration

Various types of Expansion Joints are use to take care of displacements in

a. Axial
b. Lateral
c. Angular Rotation and
d. Torsion

Proper guide spacing is essential for successful operation of Expansion Joints. The Expansion Joint Manufacturer’s Association Inc. (EJMA), a US based body, recommends a guide spacing from expansion joint based on the pipe diameter, D, as follows:

• First guide at a distance of 4D from the joint

• Second guide at a distance of 14D from first guide

If the in-built flexibility of the piping system is not adequate to take care of thermal expansion, the piping system would be subjected to stresses exceeding yield point limit, causing plastic deformation and permanent damage of the system. Expansion joint are employed in such cases. They are capable of absorbing the thermal deflections/ expansions in a manner safeguarding the piping system.

There are 3 types of expansion joints commonly used.

1. Slip type/ telescope type.
2. Corrugated type/ bellows type
3. Expansion loops

1. Slip Type/ Telescope Type

It consists of a slip-element sliding telescopically within fixed piping, serving as outer jacket. The lean… path between jacket and slip element is sealed using packing materials. Compatible with the fluid a operating temperatures.

This type of joint has following characteristics:

a) Since the leak-tightness is obtained by the packing, it results two contradictory requirements.

i. For proper leak-tightness a high tightening pressure is required.

ii. For easy movements of telescope sleeve, low tightening pressure is required.

In actual practice therefore, there is limit for applying tightening pressure and thus a perfect leak-proc….. point cannot be obtained.

Therefore it is not suitable for zero leakage operations (i.e. for toxic chemical radioactive services etc.)

b) Packing material wears out due to cyclic movement of the sleeve, causing contamination of packing material. It therefore requires periodic replacement/ maintenance and contamination makes unsuitable for
food/ pharmaceutical services.

c) Requires highly accurate alignment of piping on either side to ensure uniform tightness all around.

d) Packing materials being essentially non-metallic elements, it has its own limitations for pressure temperature applications.

e) Suitable only for axial expansion. Any angular (rotation), lateral (offset) displacement of pipes cause unequal compression of packing & hence leakage.

f) Essentially suitable for low temp. Low-pressure applications.

2. Bellows Type

Metallic bellows of compatible materials (using stainless steels) and thinner than piping thickness are to compensate the thermal expansion. Thickness of bellows is of the order of 1.0 to 2.0 mm.

Bellow type joints have several advantages over telescopic type.

a) No packing materials, hence no potential leakage points.

b) No contamination problems, no wear and tear of packing and no replacement shutdown changing the packing materials.

c) Can by used in services, which also call for some degree of angular movement or movements, in addition to axial movements, However cannot absorb torsional movements.

d) Metal bellows are thinner than piping materials, hence susceptible to rupture by over pressure.

e) Bellows can also fail in fatigue due to:

i. Stress concentration at crest or valley of corrugations.

ii. Repeated exposure to cyclic stresses.

3. Expansion Loops

Expansion loops are of same pipe material. They can absorb bending and torsion of pipe. They are most suitable for high pressure/ temperature applications. However, they occupy larger space and are heavy and bulky. Although several types of configurations are possible, the most preferred is U-type in single plane or two-plane.

Pipe Supports:

Pipe supports are provided as means to transfer to soil:

a. The load of piping system (dead load, product load)

b. Loads due to pressure- induced effects, vibrations, wind etc.

c. Transient load effects.

The piping supporting elements shall be provided in such a manner that:

1. Piping supports do not cause excessive interference with thermal expansion and contraction of pipe, which is otherwise adequately flexible.

2. They should not contribute to leakage at joints or excessive stresses at the points where they support piping system.

3. Be such that a complete release of the load will be prevented in the event of spring failures, or transient loads imposed on piping system.

Supports can be broadly classified as:

• Hanger type (Suspended from ceiling)

• Support or resting type ( on ground or structure)

• Secondary supports (Not directly attached to pipe)

Hanger rods may be with or without springs. Springs are also used in resting types.

Spring Supports

Spring supports are of two types:

• Variable springs

• Constant springs

Constant springs are used when expected displacements are of very high magnitude and/or supports are located nearby critical and sensitive equipment such as pumps, turbines etc. This is to prevent load being transferred to connected equipment after displacement.

Variable springs are used at balance applications. These can be safety used, for variation of loads below 25%. Load variation V, is defined as:

V = x 100

(Cold load – Hot load = Spring rate x travel)

Variable springs transfer the differential load (occurring due to displacement) to connected equipment or adjacent supports.

Geometric chart for components under external or compressive Loadings (for all materials)
Geometric chart for components under external or compressive Loadings (for all materials)

Figure 1.3 Chart for determining shell thickness of components under external pressure when constructed of carbon or low alloy steels (specified minimum yield strength 24,000 psi to, but not including, 30,000 psi) [Note (1)]

Figure 1.4 Chart for determining shell thickness of components under external pressure when constructed of carbon or low alloy steels (Specified minimum yield strength 30,000 psi and over except for materials
within this range where other specific charts are referenced) and type 405 and type 410 stainless steels [Note(1)]

Appendix – B

Nomenclature for miter bends
Figure 1.5 Nomenclature for miter bends

Nomenclature for fittings

Nomenclature wall thickness for fittings

Branch Connection Nomenclature

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  1. Guys : Wat about the Buried Pipeline Analysis criteria..>?

  2. If you have got anything to share regarding this topic, do let me know :)

  3. is it also applicable for tunneling? also how to compute the stresses when it is submerge underground .

  4. hey ankit all of topic is nicely elaborate..Thanks man..good help for brginer..But my problem is am going to attain a course on piping stres analysis.please send me any book related to piping stress analysis on my email, that is
    Mayur Patel

  5. please help me, any equation for find the value of y (resultant of displacement stress)? or is it considering by any table of codes?



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